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13
On the computational content of the axiom of choice
 The Journal of Symbolic Logic
, 1998
"... We present a possible computational content of the negative translation of classical analysis with the Axiom of Choice. Our interpretation seems computationally more direct than the one based on Godel's Dialectica interpretation [10, 18]. Interestingly, thisinterpretation uses a re nement of the rea ..."
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Cited by 34 (1 self)
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We present a possible computational content of the negative translation of classical analysis with the Axiom of Choice. Our interpretation seems computationally more direct than the one based on Godel's Dialectica interpretation [10, 18]. Interestingly, thisinterpretation uses a re nement of the realizibility semantics of the absurdity proposition, which is not interpreted as the empty type here. We alsoshowhow to compute witnesses from proofs in classical analysis, and how to interpret the axiom of Dependent Choice and Spector's Double Negation Shift.
On the NoCounterexample Interpretation
 J. SYMBOLIC LOGIC
, 1997
"... In [15],[16] Kreisel introduced the nocounterexample interpretation (n.c.i.) of Peano arithmetic. In particular he proved, using a complicated "substitution method (due to W. Ackermann), that for every theorem A (A prenex) of firstorder Peano arithmetic PA one can find ordinal recursive functi ..."
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Cited by 18 (10 self)
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In [15],[16] Kreisel introduced the nocounterexample interpretation (n.c.i.) of Peano arithmetic. In particular he proved, using a complicated "substitution method (due to W. Ackermann), that for every theorem A (A prenex) of firstorder Peano arithmetic PA one can find ordinal recursive functionals \Phi A of order type ! " 0 which realize the Herbrand normal form A of A. Subsequently more
Functional interpretation and inductive definitions
 Journal of Symbolic Logic
"... Abstract. Extending Gödel’s Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finitetype functionals defined using transfinite recursion on wellfounded trees. 1. ..."
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Cited by 7 (2 self)
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Abstract. Extending Gödel’s Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finitetype functionals defined using transfinite recursion on wellfounded trees. 1.
Inductive Definitions and Type Theory: An Introduction
"... MartinLof's type theory can be described as an intuitionistic theory of iterated inductive definitions developed in a framework of dependent types. It was originally intended to be a fullscale system for the formalization of constructive mathematics, but has also proved to be a powerful framewo ..."
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Cited by 6 (0 self)
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MartinLof's type theory can be described as an intuitionistic theory of iterated inductive definitions developed in a framework of dependent types. It was originally intended to be a fullscale system for the formalization of constructive mathematics, but has also proved to be a powerful framework for programming. The theory integrates an expressive specification language (its type system) and a functional programming language (where all programs terminate). There now exist several proofassistants based on type theory, and many nontrivial examples from programming, computer science, logic, and mathematics have been implemented using these. In this series of lectures we shall describe type theory as a theory of inductive definitions. We emphasize its open nature: much like in a standard functional language such as ML or Haskell the user can add new types whenever there is a need for them. We discuss the syntax and semantics of the theory. Moreover, we present some examples ...
Admissible Proof Theory And Beyond
 Logic, Methodology, and the Philosophy of Science IX, Elsevier
, 1994
"... This article will survey the state of the art nowadays, in particular recent advance in proof theory beyond admissible proof theory, giving some prospects of success of obtaining an ordinal analysis of \Pi ..."
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Cited by 5 (2 self)
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This article will survey the state of the art nowadays, in particular recent advance in proof theory beyond admissible proof theory, giving some prospects of success of obtaining an ordinal analysis of \Pi
Fragments of KripkePlatek Set Theory with Infinity
, 1992
"... In this paper we shall investigate fragments of KripkePlatek set theory with Infinity which arise from the full theory by restricting Foundation to \Pi n Foundation, where n 2. The strength of such fragments will be characterized in terms of the smallest ordinal ff such that L ff is a model o ..."
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Cited by 4 (4 self)
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In this paper we shall investigate fragments of KripkePlatek set theory with Infinity which arise from the full theory by restricting Foundation to \Pi n Foundation, where n 2. The strength of such fragments will be characterized in terms of the smallest ordinal ff such that L ff is a model of every \Pi 2 sentence which is provable in the theory. 1 Introduction KripkePlatek set theory plus Infinity (hereinafter called KP!) is a truly remarkable subsystem of ZF. Though considerably weaker than ZF, a great deal of set theory requires only the axioms of this subsystem (cf.[Ba]). KP! consists of the axioms Extensionality, Pair, Union, (Set)Foundation, Infinity, along with the schemas of \Delta 0 Collection, \Delta 0 Separation, and Foundation for Definable Classes. So KP! arises from ZF by completely omitting Power Set and restricting Separation and Collection to \Delta 0 formulas. These alterations are suggested by the informal notion of "predicative". KP! is an impredicat...
THE BOUNDED FUNCTIONAL INTERPRETATION OF THE DOUBLE NEGATION SHIFT
"... Abstract. We prove that the (nonintuitionistic) law of the double negation shift has a bounded functional interpretation with bar recursive functionals of finite type. As an application, we show that full numerical comprehension is compatible with the uniformities introduced by the characteristic p ..."
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Abstract. We prove that the (nonintuitionistic) law of the double negation shift has a bounded functional interpretation with bar recursive functionals of finite type. As an application, we show that full numerical comprehension is compatible with the uniformities introduced by the characteristic principles of the bounded functional interpretation for the classical case. §1. Introduction and background. In 1962 [14], Clifford Spector gave a remarkable characterization of the provably recursive functionals of full secondorder arithmetic (a.k.a. analysis). The central result of his paper is an extension, from arithmetic to analysis, of the (then quite recent) dialectica interpretation of Gödel of 1958 [7]. Spector’s extension relies on a form of wellfounded recursion
Gödel's Dialectica interpretation and its twoway stretch
 in Computational Logic and Proof Theory (G. Gottlob et al eds.), Lecture Notes in Computer Science 713
, 1997
"... this article has appeared in Computational Logic and Proof Theory (Proc. 3 ..."
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this article has appeared in Computational Logic and Proof Theory (Proc. 3
KripkePlatek Set Theory And The AntiFoundation Axiom
"... . The paper investigates the strength of the AntiFoundation Axiom, AFA, on the basis of KripkePlatek set theory without Foundation. It is shown that the addition of AFA considerably increases the proof theoretic strength. MSC:03F15,03F35 Keywords: Antifoundation axiom, KripkePlate set theory ..."
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. The paper investigates the strength of the AntiFoundation Axiom, AFA, on the basis of KripkePlatek set theory without Foundation. It is shown that the addition of AFA considerably increases the proof theoretic strength. MSC:03F15,03F35 Keywords: Antifoundation axiom, KripkePlate set theory, subsystems of second order arithmeic 1. Introduction Intrinsically circular phenomena have come to the attention of researchers in differing fields such as mathematical logic, computer science, artificial intelligence, linguistics, cognitive science, and philosophy. Logicians first explored set theories whose universe contains what are called nonwellfounded sets, or hypersets (cf. [6], [2]). But the area was considered rather exotic until these theories were put to use in developing rigorous accounts of circular notions in computer science (cf. [4]). Instead of the Foundation Axiom these set theories adopt the socalled AntiFoundation Axiom, AFA, which gives rise to a rich universe of ...